In classical mechanics and solid-state physics texts, you often see the Hamiltonian $H$ of a system of harmonic oscillators written in normal mode coordinates:
$$ H = \frac{1}{2}\sum_n\dot{X_n}^2 + \frac{1}{2}\sum_n\omega_n^2X_n^2$$
where $X_n$ and $\dot{X_n}$ are the amplitudes velocities of mode $n$. In the potential energy term, $\omega_n$ is the angular frequency of mode $n$, because $\omega_n$ comes from the eigenvalue equation for normal modes. Let's look at the units of this Hamiltonian.
Consider SI units:
- $\omega_n = rad/s$
- $X_n = \sqrt{kg}m$
- $\dot{X_n} = \sqrt{kg}m/s$
- $\dot{X_n}^2 = J$
- $\omega_n^2X_n^2 = rad^2J$
Shouldn't the potential energy have units of $J$ instead of $rad^2J$? Do we need to divide by $(2\pi)^2$?
In my experience of calculating mode energies in molecular dynamics simulations, the use of angular frequency gives the correct energy, but the units don't make sense!
